Finding Pre-Image Coordinates Under Transformation R_y=-x
In the realm of coordinate geometry, transformations play a crucial role in mapping points and figures from one location to another. Among these transformations, the reflection across the line y = -x holds a unique position. This particular transformation, denoted as r_y=-x, swaps the x and y coordinates of a point and negates them. In simpler terms, if a point has coordinates (x, y), its image after the transformation will have coordinates (-y, -x). Understanding this transformation is key to solving problems involving pre-images and images of points.
Delving Deeper into Reflections: Reflections, in general, are transformations that mirror a point or figure across a line, known as the line of reflection. The line y = -x serves as our line of reflection in this case. To visualize this, imagine folding the coordinate plane along the line y = -x. The reflected image of a point will land exactly where its pre-image was before the fold. The transformation r_y=-x is a specific type of reflection with distinct properties. It not only reflects the point across the line but also inverts the signs of both coordinates. This characteristic is vital for accurately determining the pre-image of a point.
Mathematical Representation: The transformation r_y=-x can be formally expressed as a mapping rule: (x, y) → (-y, -x). This rule provides a concise way to understand how the coordinates of a point change under this transformation. The x-coordinate of the pre-image becomes the negated y-coordinate of the image, and the y-coordinate of the pre-image becomes the negated x-coordinate of the image. This swapping and negation are the essence of the r_y=-x transformation. Mastering this rule is essential for solving problems involving transformations and coordinate geometry. The ability to mentally apply this rule or write it down explicitly can significantly simplify the process of finding pre-images and images.
Given the image of a point as (-4, 9) after the transformation r_y=-x, our objective is to find the coordinates of its pre-image. To achieve this, we need to reverse the transformation process. Remember, the transformation r_y=-x maps a point (x, y) to (-y, -x). Therefore, to find the pre-image, we need to reverse this mapping. This involves negating the coordinates of the image and swapping them back to their original positions.
Reversing the Transformation: If the image is (-4, 9), we can denote the pre-image as (x, y). Applying the transformation r_y=-x, we have (x, y) → (-y, -x). This means that (-y, -x) = (-4, 9). Now, we can set up two simple equations: -y = -4 and -x = 9. Solving these equations will give us the coordinates of the pre-image. The first equation, -y = -4, implies that y = 4. The second equation, -x = 9, implies that x = -9. Therefore, the pre-image has coordinates (-9, 4).
Step-by-Step Solution: Let's break down the process into clear steps:
- Identify the image: The image is given as (-4, 9).
- Apply the reverse transformation: To reverse the transformation r_y=-x, we negate the coordinates and swap them. So, we have (-4, 9) → (-9, -(-4)) → (-9, 4).
- The pre-image: The coordinates of the pre-image are (-9, 4).
This step-by-step approach ensures clarity and accuracy in finding the pre-image. It highlights the importance of understanding the reverse transformation and applying it systematically.
To ensure the accuracy of our solution, we can apply the transformation r_y=-x to the pre-image we found, which is (-9, 4). If the resulting image matches the given image (-4, 9), our solution is correct. Applying the transformation r_y=-x to (-9, 4), we swap the coordinates and negate them:
(-9, 4) → (-4, -(-9)) → (-4, 9).
Confirming the Result: The resulting image after applying the transformation to our calculated pre-image is indeed (-4, 9), which matches the given image. This confirms that our solution for the pre-image coordinates is correct. Verifying the solution is a crucial step in problem-solving, especially in mathematics. It helps to catch any potential errors and reinforces the understanding of the concepts involved. By applying the original transformation to the pre-image and checking if it results in the given image, we gain confidence in our answer.
Importance of Verification: Verification is not just a formality; it's an integral part of the problem-solving process. It provides a safety net, ensuring that the final answer is accurate. In complex problems, errors can easily occur during the intermediate steps. Verifying the solution helps to identify and correct these errors. Furthermore, the act of verification deepens the understanding of the underlying concepts and the relationships between different elements of the problem.
In conclusion, given the image (-4, 9) and the transformation rule r_y=-x, the coordinates of the pre-image are (-9, 4). This was determined by understanding the transformation r_y=-x, which swaps the x and y coordinates and negates them, and then reversing this process to find the pre-image. The solution was further verified by applying the transformation to the pre-image and confirming that it resulted in the given image. The correct answer is therefore A. (-9, 4).
Recap of Key Concepts: Let's recap the key concepts involved in solving this problem:
- Transformation r_y=-x: This transformation reflects a point across the line y = -x, swapping the x and y coordinates and negating them.
- Pre-image and Image: The pre-image is the original point, and the image is the point after the transformation.
- Reversing the Transformation: To find the pre-image, we need to reverse the transformation process.
- Verification: It is crucial to verify the solution by applying the transformation to the pre-image and confirming that it results in the given image.
By mastering these concepts, you can confidently solve similar problems involving transformations and coordinate geometry. Remember to practice and apply these concepts in various contexts to solidify your understanding.
